Scientia Magna Vol. 2 (2006), No. 1, 30-34
A note on q-nanlogue of S´andor’s functions Taekyun Kim Department of Mathematics Education Kongju National University Kongju 314- 701 , South Korea C. Adiga and Jung Hun Han Department of Studies in Mathematics University of Mysore Manasagangotri Mysore 570006, India Abstract The additive analogues of Pseudo-Smarandache, Smarandache-simple functions and their duals have been recently studied by J. S´ andor. In this note, we obtain q-analogues of S´ andor’s theorems [6]. Keywords q-gamma function; Pseudo-Smarandache function; Smarandache-simple function; Asymtotic formula.
Dedicated to Sun-Yi Park on 90th birthday
§1. Introduction The additive analogues of Smarandache functions S and S∗ have been introduced by S´ andor [5] as follows: S(x) = min{m ∈ N : x ≤ m!},
x ∈ (1, ∞),
S∗ (x) = max{m ∈ N : m! ≤ x},
x ∈ [1, ∞),
and
He has studied many important properties of S∗ relating to continuity, differentiability and Riemann integrability and also p roved the following theorems: Theorem 1.1. S∗ ∼ Theorem 1.2.
The series
log x log log x ∞ X
(x → ∞).
1 , n(S∗ (n))α n=1 is convergent for α > 1 and divergent for α ≤ 1.
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A note on q-nanlogue of S´ andor’s functions
In [1], Adiga and Kim have obtained generalizations of Theorems 1.1 and 1.2 by the use of Euler’s gamma function. Recently Adiga-Kim-Somashekara-Fathima [2] have established a qanalogues of these results on employing analogues of Pseudo-Smarandache, Smarandache-simple functions and their duals as follows: ½ ¾ m(m + 1) Z(x) = min m ∈ N : x ≤ , 2
x ∈ (0, ∞),
½ ¾ m(m + 1) Z∗ (x) = max m ∈ N : ≤x , 2
x ∈ [1, ∞),
P (x) = min{m ∈ N : px ≤ m!},
p > 1,
P∗ (x) = max{m ∈ N : m! ≤ px }, He has also proved the following theorems: Theorem 1.3. 1√ Z∗ ∼ 8x + 1 2 Theorem 1.4. The series ∞ X 1 n=1
x ∈ (0, ∞),
p > 1,
x ∈ [1, ∞).
(x → ∞).
(Z∗ (n))α
,
is convergent for α > 2 and divergent for α ≤ 2. The series ∞ X
1 , n(Z∗ (n))α n=1 is convergent for all α > 0. Theorem 1.5. log P∗ (x) ∼ log x (x → ∞), Theorem 1.6.
The series
µ ¶α ∞ X 1 log log n n log P∗ (n) n=1
is convergent for all α > 1 and divergent for α ≤ 1. The main purpose of this note is to obtain q-analogues of S´ andor’s Theorems 1.3 and 1.5. In what follows, we make u se of the following notations and definitions. F. H. Jackson defined a q-analogues of the gamma function which extends the q-factorial (n!)q = 1(1 + q)(1 + q + q 2 ) · · · (1 + q + q 2 + · · · + q n−1 ),
cf
[3],
which becomes the ordinary factorial as q → 1. He defined the q-analogue of the gamma function as Γq (x) =
(q; q)∞ (1 − q)1−x , (q x ; q)∞
0 < q < 1,
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T. Kim, C. Adiga and J. H. Han
No. 1
and Γq (x) =
x (q −1 ; q −1 )∞ (1 − q)1−x q (2) , q > 1, (q −x ; q −1 )∞
where (a; q)∞ =
∞ Y
(1 − aq n ).
n=0
It is well known that Γq (x) → Γ(x) as q → 1, where Γ(x) is the ordinary gamma function.
§2. Main Theorems We now defined the q-analogues of Z and Z∗ as follows: ½ ¾ 1 − qm Γq (m + 2) Zq (x) = min :x≤ , m ∈ N, 1−q 2Γq (m) and
½ Zq∗ (x)
= max
¾ 1 − q m Γq (m + 2) : ≤x , 1−q 2Γq (m)
x ∈ (0, ∞),
·
m ∈ N,
¶ Γq (m + 2) x∈ ,∞ , 2Γq (1)
where 0 < q < 1. Clearly, Zq (x) → Z(x) and Zq∗ (x) → Z∗ (x) as q → 1− . From the definitions of Zq and Zq∗ , it is clear that ³ i Γ (3) 1, if x ∈ 0, 2Γqq (1) ³ i Zq (x) = (1) 1−qm , ifx ∈ Γq (m+1) , Γq (m+2) , m ≥ 2, 1−q 2Γq (m−1) 2Γq (m) and Zq∗ =
1 − qm 1−q
· if x ∈
Γq (m + 2) Γq (m + 3) , 2Γq (m) 2Γq (m + 1)
¶ .
(2)
Since 1 − q m−1 1 − qm 1 − q m−1 1 − q m−1 ≤ = + q m−1 ≤ + 1, 1−q 1−q 1−q 1−q (1) and (2) imply that for x >
Γq (3) 2Γq (1) ,
Zq∗ ≤ Zq ≤ Zq∗ + 1. Hence it suffices to study the function Zq∗ . We now prove our main theorems. Theorem 2.1. If 0 < q < 1, then √ √ 1 + 8xq − (1 + 2q) 1 + 8xq − 1 Γq (3) ∗ < Zq ≤ , x≥ . 2 2q 2q 2Γq (1) Proof. If Γq (k + 2) Γq (k + 3) ≤x< , 2Γq (k) 2Γq (k + 1)
(3)
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A note on q-nanlogue of S´ andor’s functions
then Zq∗ =
1 − qk 1−q
and (1 − q k )(1 − q k+1 ) − 2x(1 − q)2 ≤ 0 < (1 − q k+1 )(1 − q k+2 ) − 2x(1 − q)2 .
(4)
Consider the functions f and g defined by f (y) = (1 − y)(1 − yq) − 2x(1 − q)2 and g(y) = (1 − yq)(1 − yq 2 ) − 2x(1 − q)2 . Note that f is monotonically decreasing for y ≤ Also f (y1 ) = 0 = g(y2 ) where
1+q 2q
and g is strictly decreasing for y ≤
1+q 2q 2 .
√ (1 + q) − (1 − q) 1 + 8xq , 2q √ (q + q 2 ) − q(1 − q) 1 + 8xq . y2 = 2q 3 y1 =
Since y1 ≤
1+q 2q ,
y2 ≤
1+q 2q 2
and q k <
1+q 2q
<
1+q 2q 2 ,
from (4), it follows that
f (q k ) ≤ f (y1 ) = 0 = g(y2 ) < g(q k ). Thus y1 < q k < y2 and hence 1 − qk 1 − y1 1 − y2 < < . 1−q 1−q 1−q i.e.
√
1 + 8xq − (1 + 2q) < Zq∗ ≤ 2q 2
√
1 + 8xq − 1 . 2q
This completes the proof. Remark. Letting q − 1− in the above theorem, we obtain S´ andor’s Theorem 1.3. We define the q-analogues of P and P∗ as follows: Pq (x) = min{m ∈ N : px ≤ Γq (m + 1)},
p > 1,
x ∈ (0, ∞),
Pq∗ (x) = max{m ∈ N : Γq (m + 1) ≤ px },
p > 1,
x ∈ [1, ∞),
and where 0 < q < 1. Clearly, Pq (x) → P (x) and Pq∗ → P∗ (x) as q → 1− . From the definitions of Pq and Pq∗ , we have Pq∗ (x) ≤ Pq (x) ≤ Pq∗ (x) + 1. Hence it is enough to study the function Pq∗ . Theorem 2.2. If 0 < q < 1, then P∗ (x) ∼
x log p ´ ³ 1 log 1−q
(x → ∞).
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No. 1
Proof. If Γq (n + 1) ≤ px < Γq (n + 2), then Pq∗ (x) = n and log Γq (n + 1) ≤ log px < log Γq (n + 2). But by the q-analogue of Stirling’s formula established by Moak [4], we have µ ¶ µ n+1 ¶ µ ¶ 1 q 1 log Γq (n + 1) ∼ n + log ∼ n log . 2 q−1 1−q ³ ´ 1 Dividing (5) throughout by n log 1−q , we obtain log Γq (n + 1) x log p log Γq (n + 2) ³ ´ ≤ ´< ³ ´. ³ 1 1 1 n log 1−q Pq∗ (x) log 1−q n log 1−q
(5)
(6)
(7)
Using (6) and (7), we deduce lim
x→∞
x log p ³ Pq∗ (x) log
1 1−q
´ = 1.
This completes the proof.
References [1] C. Adiga and T. Kim, On a generalization of S´ andor’s function, Proc. Jangjeon Math. Soc., 5(2002), 121-124. [2] C. Adiga, T. Kim, D. D. Somashekara and N. Fathima, On a q-analogue of S´ andor’s function, J. Inequal. Pure and Appl. Math., 4(2003), 1-5. [3] T. Kim, Non-archimedean q-integrals with multiple Changhee q-Bernoulli polynomials, Russian J. Math. Phys., 10(2003), 91-98. [4] D. S. Moak, The q-analogue of Stirling’s formula, Rocky Mountain J. Math., 14(1984), 403-413. [5] J. S´ andor, On an additive analogue of the function S, Note Numb. Th. Discr. Math., 7(2001), 91-95. [6] J. S´ andor, On an additive analogue of certain arithmetic function, J. Smarandache Notions, 14(2004), 128-133.